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Theorem inawina 9141
Description: Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
inawina  |-  ( A  e.  Inacc  ->  A  e.  InaccW )

Proof of Theorem inawina
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfon 8711 . . . . 5  |-  ( cf `  A )  e.  On
2 eleq1 2528 . . . . 5  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  On  <->  A  e.  On ) )
31, 2mpbii 216 . . . 4  |-  ( ( cf `  A )  =  A  ->  A  e.  On )
433ad2ant2 1036 . . 3  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  A  e.  On )
5 idd 25 . . . 4  |-  ( A  e.  On  ->  ( A  =/=  (/)  ->  A  =/=  (/) ) )
6 idd 25 . . . 4  |-  ( A  e.  On  ->  (
( cf `  A
)  =  A  -> 
( cf `  A
)  =  A ) )
7 inawinalem 9140 . . . 4  |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A  ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
85, 6, 73anim123d 1355 . . 3  |-  ( A  e.  On  ->  (
( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) ) )
94, 8mpcom 37 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
10 elina 9138 . 2  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
11 elwina 9137 . 2  |-  ( A  e.  InaccW  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
129, 10, 113imtr4i 274 1  |-  ( A  e.  Inacc  ->  A  e.  InaccW )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   (/)c0 3743   ~Pcpw 3963   class class class wbr 4416   Oncon0 5442   ` cfv 5601    ~< csdm 7594   cfccf 8397   InaccWcwina 9133   Inacccina 9134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-wrecs 7054  df-recs 7116  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-card 8399  df-cf 8401  df-wina 9135  df-ina 9136
This theorem is referenced by:  gchina  9150  inar1  9226  inatsk  9229  tskuni  9234  grur1a  9270  grur1  9271  inaprc  9287
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